ODBC settings are on each client computer. Go to Control Panel / Administrative Tools, and there should be an option called Data Sources (ODBC). From there, you should be able to find the DSN that the client program is using, and looking at the DSN (I believe it's the configure button) will tell you how the client is connecting to the server, what user it is, etc.

If it's using Windows Authentication, then the connection to the database is dependent on what network user is logged into the client computer. You would have to verify in Management Studio that there is a SQL Server login setup for that network user, and that login has a user with privileges in the specific database.